Expected value

The mean or expected value of a discrete random variable, symmetric distributions, fair games and expected gain, and the expected value of a linear function aX + b.

The mean

Expected value

$$\mu = E(X) = \sum x\,P(X = x)$$

It is the long-run average value of $X$ over many trials. It need not be a value that $X$ can actually take.

Example 1: a die●

Find the expected score on a fair die.

$E(X) = (1 + 2 + 3 + 4 + 5 + 6) \times \tfrac16 = 3.5$

Example 2●●

Find $E(X)$ for this distribution.

$x$	1	2	3	4
$P(X = x)$	$0.1$	$0.3$	$0.4$	$0.2$

$E(X) = 0.1 + 0.6 + 1.2 + 0.8 = 2.7$

Games and expected gain

Example 3●●

A game costs $\$5$. You win $\$20$ with probability $0.2$, and otherwise nothing. Is the game fair?

Expected winnings $= 20 \times 0.2 = \$4$.

Expected gain $= 4 - 5 = -\$1$ per game. It is not fair: on average a player loses $\$1$ a game.

A game is fair when the expected gain is $0$.

Linear changes

Mean of aX + b

$$E(aX + b) = aE(X) + b$$

Example 4●

$E(X) = 2.7$. Find $E(10X - 5)$.

$10 \times 2.7 - 5 = 22$

Flashcards

$E(X)$

$\sum x\,P(X = x)$

Fair game

Expected gain is $0$.

$E(aX + b)$

$aE(X) + b$

Practice

Find the expected score on a fair die.
Find $E(X)$ in Example 2.
A game costs $\$5$, and you win $\$20$ with probability $0.2$. Find the expected gain in dollars.
What should the game cost to be fair?
$E(X) = 2.7$. Find $E(10X - 5)$.
Two fair coins are tossed. Find the expected number of heads.
The expected value of a random variable:
$E(X) = 3.5$ for a die. Find $E(2X)$.
A game is fair when: